-\title{Energy Consumption Reduction with DVFS for Message \\
- Passing Iterative Applications on \\
- Grid Architecture}
+\title{Optimizing Energy Consumption with DVFS for Message \\
+ Passing Applications \textcolor{blue}{with iterations} on \\
+ Grid Architectures}
In this paper, a new online frequency selecting algorithm for grids, composed of heterogeneous clusters, is presented.
It selects the frequencies and tries to give the best
trade-off between energy saving and performance degradation, for each node
- computing the message passing iterative application.
+ computing the message passing application \textcolor{blue}{with iterations}.
The algorithm has a small
overhead and works without training or profiling. It uses a new energy model
- for message passing iterative applications running on a grid.
+ for message passing applications \textcolor{blue}{with iterations} running on a grid.
The proposed algorithm is evaluated on a real grid, the Grid'5000 platform, while
running the NAS parallel benchmarks. The experiments on 16 nodes, distributed on three clusters, show that it reduces on average the
energy consumption by \np[\%]{30} while the performance is on average only degraded
50.32 kilowatts.
Besides platform improvements, there are many software and hardware techniques
-to lower the energy consumption of these platforms, such as scheduling, DVFS,
-\dots{} DVFS is a widely used process to reduce the energy consumption of a
+to lower the energy consumption of these platforms, such as DVFS, scheduling \textcolor{blue}{and other techniques}.
+ DVFS is a widely used process to reduce the energy consumption of a
processor by lowering its frequency
\cite{Rizvandi_Some.Observations.on.Optimal.Frequency}. However, it also reduces
the number of FLOPS executed by the processor which may increase the execution
time of the application running over that processor. Therefore, researchers use
different optimization strategies to select the frequency that gives the best
trade-off between the energy reduction and performance degradation ratio. In
-\cite{Our_first_paper} and \cite{pdsec2015} , a frequency selecting algorithm
-was proposed to reduce the energy consumption of message passing iterative
-applications running over homogeneous and heterogeneous clusters respectively.
+\cite{Our_first_paper} and \cite{pdsec2015}, a frequency selecting algorithm
+was proposed to reduce the energy consumption of message passing
+applications \textcolor{blue}{with iterations} running over homogeneous and heterogeneous clusters respectively.
The results of the experiments showed significant energy consumption
reductions. All the experimental results were conducted over the SimGrid
-simulator \cite{SimGrid}, which offers easy tools to create homogeneous and
-heterogeneous platforms and runs message passing parallel applications over
-them. %
-\AG{[\dots], which offers easy tools to describe homogeneous and heterogeneous
- platforms, and to simulate the execution of message passing parallel
- applications over them.}%
+simulator \cite{SimGrid}, which offers easy tools to describe homogeneous and heterogeneous platforms, and to simulate the execution of message passing parallel
+applications over them.
+
In this paper, a new frequency selecting algorithm, adapted to grid platforms
composed of heterogeneous clusters, is presented. It is applied to the NAS
parallel benchmarks and evaluated over a real testbed, the Grid'5000 platform
\cite{grid5000}. It selects for a grid platform running a message passing
-iterative application the vector of frequencies that simultaneously tries to
+ application \textcolor{blue}{with iterations} the vector of frequencies that simultaneously tries to
offer the maximum energy reduction and minimum performance degradation
ratios. The algorithm has a very small overhead, works online and does not need
any training or profiling.
Section~\ref{sec.optim} details the proposed frequencies selecting algorithm.
Section~\ref{sec.expe} presents the results of applying the algorithm on the
NAS parallel benchmarks and executing them on the Grid'5000 testbed.
-It also evaluates the algorithm over multi-cores per node architectures and over three different power scenarios. Moreover, it shows the
+It also evaluates the algorithm over multi-core per node architectures and over three different power scenarios. Moreover, it shows the
comparison results between the proposed method and an existing method. Finally,
in Section~\ref{sec.concl} the paper ends with a summary and some future works.
+
\section{Related works}
\label{sec.relwork}
varied objectives such as energy reduction or performance. They also could be
adapted to the execution's environment and the type of the application such as
sequential, parallel or distributed architecture, homogeneous or heterogeneous
-platform, synchronous or asynchronous application, \dots{}
+platform, synchronous or asynchronous application.
In this paper, we are interested in reducing energy for message passing
-iterative synchronous applications running over heterogeneous grid platforms. Some
+ synchronous applications \textcolor{blue}{with iterations} running over heterogeneous grid platforms. Some
works have already been done for such platforms and they can be classified into
two types of heterogeneous platforms:
\begin{itemize}
overhead. In contrast to the above described papers, this paper presents the
following contributions :
\begin{enumerate}
-\item two new energy and performance models for message passing iterative
- synchronous applications running over a heterogeneous grid platform. Both models
+\item two new energy and performance models for message passing
+ synchronous applications \textcolor{blue}{with iterations} running over a heterogeneous grid platform. Both models
take into account communication and slack times. The models can predict the
required energy and the execution time of the application.
platforms. The algorithm has a very small overhead and does not need any
training nor profiling. It uses a new optimization function which
simultaneously maximizes the performance and minimizes the energy consumption
- of a message passing iterative synchronous application.
+ of a message passing synchronous application \textcolor{blue}{with iterations}.
\end{enumerate}
\section{The performance and energy consumption measurements on heterogeneous grid architecture}
\label{sec.exe}
-\subsection{The execution time of message passing distributed iterative
- applications on a heterogeneous platform}
+\subsection{The execution time of message passing distributed
+ applications \textcolor{blue}{with iterations} on a heterogeneous platform}
In this paper, we are interested in reducing the energy consumption of message
-passing distributed iterative synchronous applications running over
+passing distributed synchronous applications \textcolor{blue}{with iterations} running over
heterogeneous grid platforms. A heterogeneous grid platform could be defined as a collection of
heterogeneous computing clusters interconnected via a long distance network which has lower bandwidth
and higher latency than the local networks of the clusters. Each computing cluster in the grid is composed of homogeneous nodes that are connected together via high speed network. Therefore, each cluster has different characteristics such as computing power (FLOPS), energy consumption, CPU's frequency range, network bandwidth and latency.
-The overall execution time of a distributed iterative synchronous application
+The overall execution time of a distributed synchronous application \textcolor{blue}{with iterations}
over a heterogeneous grid consists of the sum of the computation time and
-the communication time for every iteration on a node. However, due to the
-heterogeneous computation power of the computing clusters, slack times may occur
-when fast nodes have to wait, during synchronous communications, for the slower
-nodes to finish their computations (see Figure~\ref{fig:heter}). Therefore, the
+the communication time for every iteration on a node.
+\textcolor{blue}{However, nodes from distinct clusters in a grid have different computing powers, thus
+while executing message passing \textcolor{blue}{with iterations} synchronous applications, fast nodes
+have to wait for the slower ones to finish their computations before being able
+to synchronously communicate with them as in Figure~\ref{fig:heter}. These
+periods are called idle or slack times. }
+Therefore, the
overall execution time of the program is the execution time of the slowest task
-which has the highest computation time and no slack time.
+which has the highest computation time and no slack time. \textcolor{blue}{For example, in Figure \ref{fig:heter} the task 1 is the slower task which has no slack time (not waits for the other nodes) and it is only has the communication times.}
\begin{figure}[!t]
\centering
\label{eq:s}
S = \frac{\Fmax}{\Fnew}
\end{equation}
+\textcolor{blue}{Where $\Fmax$ is the maximum frequency before applying DVFS and $\Fnew$ is the new frequency after applying DVFS.}
The execution time of a compute bound sequential program is linearly
proportional to the frequency scaling factor $S$. On the other hand, message
passing distributed applications consist of two parts: computation and
Since in a heterogeneous grid each cluster has different characteristics,
especially different frequency gears, when applying DVFS operations on the nodes
of these clusters, they may get different scaling factors represented by a scaling vector:
-$(S_{11}, S_{12},\dots, S_{NM})$ where $S_{ij}$ is the scaling factor of processor $j$ in cluster $i$ . To
-be able to predict the execution time of message passing synchronous iterative
-applications running over a heterogeneous grid, for different vectors of
+$(S_{11}, S_{12},\dots, S_{NM_i})$ where $S_{ij}$ is the scaling factor of processor $j$ in cluster $i$ . To
+be able to predict the execution time of message passing synchronous
+applications \textcolor{blue}{with iterations} running over a heterogeneous grid, for different vectors of
scaling factors, the communication time and the computation time for all the
tasks must be measured during the first iteration before applying any DVFS
operation. Then the execution time for one iteration of the application with any
-vector of scaling factors can be predicted using (\ref{eq:perf}).
+vector of scaling factors can be predicted using Equation (\ref{eq:perf}).
%
\begin{equation}
\label{eq:perf}
- \Tnew = \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}({\TcpOld[ij]} \cdot S_{ij})
- +\mathop{\min_{j=1,\dots,M}} (\Tcm[hj])
+ \Tnew = \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}({\TcpOld[ij]} \cdot S_{ij})
+ +\mathop{\min_{j=1,\dots,M_i}} (\Tcm[hj])
\end{equation}
%
-where $N$ is the number of clusters in the grid, $M$ is the number of nodes in
-each cluster, $\TcpOld[ij]$ is the computation time of processor $j$ in the cluster $i$
+where $N$ is the number of clusters in the grid, $M_i$ is the number of nodes in
+ cluster $i$, $\TcpOld[ij]$ is the computation time of processor $j$ in the cluster $i$
and $\Tcm[hj]$ is the communication time of processor $j$ in the cluster $h$ during the
first iteration. The execution time for one iteration is equal to the sum of the maximum computation time for all nodes with the new scaling factors
-and the slowest communication time without slack time during one iteration.
-The latter is equal to the communication time of the slowest node in the slowest cluster $h$.
-It means\AG[]{It means that\dots} only the communication time without any slack time is taken into account.
-Therefore, the execution time of the iterative application is equal to
-the execution time of one iteration as in (\ref{eq:perf}) multiplied by the
+and \textcolor{blue}{the communication time of the slower node without slack time during one iteration.
+The slower node $h$ is the node that gives maximum execution time in all clusters befor scaling its frequency.}
+It means that only the communication time without any slack time is taken into account.
+Therefore, the execution time of the application \textcolor{blue}{with iterations} is equal to
+the execution time of one iteration as in Equation (\ref{eq:perf}) multiplied by the
number of iterations of that application.
This prediction model is developed from the model to predict the execution time
of message passing distributed applications for homogeneous and heterogeneous clusters
-~\cite{Our_first_paper,pdsec2015}. The execution time prediction model is
-used in the method to optimize both the energy consumption and the performance
-of iterative methods, which is presented in the following sections.
+~\cite{Our_first_paper,pdsec2015}. \textcolor{blue}{where the homogeneous cluster predication model was used one scaling factor denoted as $S$, because all the nodes in the cluster have the same computing powers. Whereas, in heterogeneous cluster prediction model all the nodes have different scales and the scaling factors have denoted as one dimensional vector $(S_1, S_2, \dots, S_N)$. The execution time prediction model for a grid Equation \ref{eq:perf} defines a two dimensional array of scales
+$(S_{11}, S_{12},\dots, S_{NM_i})$}. This model is used in the method to optimize both the energy consumption and the performance of iterative methods, which is presented in the following sections.
\subsection{Energy model for heterogeneous grid platform}
\end{equation}
The static power is related to the power leakage of the CPU and is consumed
during computation and even when idle. As
-in~\cite{Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling},
+in~\cite{Rauber_Analytical.Modeling.for.Energy, Zhuo_Energy.efficient.Dynamic.Task.Scheduling},
the static power of a processor is considered as constant during idle and
computation periods, and for all its available frequencies. The static energy
is the static power multiplied by the execution time of the program. According
In the considered heterogeneous grid platform, each node $j$ in cluster $i$ may have
different dynamic and static powers from the nodes of the other clusters,
-noted as $\Pd[ij]$ and $\Ps[ij]$ respectively. Therefore, even if the distributed
-message passing iterative application is load balanced, the computation time of each CPU $j$
-in cluster $i$ noted $\Tcp[ij]$ may be different and different frequency scaling factors may be
+noted as $\Pd[ij]$ and $\Ps[ij]$ respectively. \textcolor{blue}{Therefore, even if the distributed
+message passing application \textcolor{blue}{with iterations} is load balanced, the computation time of each CPU $j$
+in cluster $i$ noted $\Tcp[ij]$ may be slightly different due to the delay caused by the scheduler of the operating system}. Therefore, different frequency scaling factors may be
computed in order to decrease the overall energy consumption of the application
-and reduce slack times. The communication time of a processor $j$ in cluster $i$ is noted as
+and reduce the slack times. The communication time of a processor $j$ in cluster $i$ is noted as
$\Tcm[ij]$ and could contain slack times when communicating with slower nodes,
see Figure~\ref{fig:heter}. Therefore, all nodes do not have equal
communication times. While the dynamic energy is computed according to the
frequency scaling factor and the dynamic power of each node as in
(\ref{eq:Edyn}), the static energy is computed as the sum of the execution time
-of one iteration multiplied by the static power of each processor. The overall
-energy consumption of a message passing distributed application executed over a
+of one iteration multiplied by the static power of each processor.
+\textcolor{blue}{ The CPU during the communication times consumes only the static power. While
+in the computation times, it consumes both the dynamic and the static power refer to \cite{Freeh_Exploring.the.Energy.Time.Tradeoff}.}
+The overall energy consumption of a message passing distributed application executed over a
heterogeneous grid platform during one iteration is the summation of all dynamic and
-static energies for $M$ processors in $N$ clusters. It is computed as follows:
+static energies for $M_i$ processors in $N$ clusters. It is computed as follows:
\begin{multline}
\label{eq:energy}
- E = \sum_{i=1}^{N} \sum_{i=1}^{M} {(S_{ij}^{-2} \cdot \Pd[ij] \cdot \Tcp[ij])} +
- \sum_{i=1}^{N} \sum_{j=1}^{M} (\Ps[ij] \cdot {} \\
- (\mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}({\Tcp[ij]} \cdot S_{ij})
- +\mathop{\min_{j=1,\dots M}} (\Tcm[hj]) ))
+ E = \sum_{i=1}^{N} \sum_{i=1}^{M_i} {(S_{ij}^{-2} \cdot \Pd[ij] \cdot \Tcp[ij])} +
+ \sum_{i=1}^{N} \sum_{j=1}^{M_i} (\Ps[ij] \cdot {} \\
+ (\mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}({\Tcp[ij]} \cdot S_{ij})
+ +\mathop{\min_{j=1,\dots M_i}} (\Tcm[hj]) ))
\end{multline}
Reducing the frequencies of the processors according to the vector of scaling
-factors $(S_{11}, S_{12},\dots, S_{NM})$ may degrade the performance of the application
+factors $(S_{11}, S_{12},\dots, S_{NM_i})$ may degrade the performance of the application
and thus, increase the static energy because the execution time is
increased~\cite{Kim_Leakage.Current.Moore.Law}. The overall energy consumption
-for the iterative application can be measured by measuring the energy
+for the application \textcolor{blue}{with iterations} can be measured by measuring the energy
consumption for one iteration as in (\ref{eq:energy}) multiplied by the number
of iterations of that application.
frequency scaling factors for a homogeneous and a heterogeneous cluster respectively, were proposed.
Both methods selects the frequencies that gives the best trade-off between
energy consumption reduction and performance for message passing
-iterative synchronous applications. In this work we
-are interested in grids that are composed of heterogeneous clusters were the nodes have different characteristics such as dynamic power, static power, computation power, frequencies range, network latency and bandwidth.
-Due to the
-heterogeneity of the processors, a vector of scaling factors should be selected
+ synchronous applications \textcolor{blue}{with iterations}. In this work we
+are interested in grids that are composed of heterogeneous clusters, \textcolor{blue}{where} the nodes
+have different characteristics such as dynamic power, static power, computation power,
+frequencies range, network latency and bandwidth.
+Due to the heterogeneity of the processors, a vector of scaling factors should be selected
and it must give the best trade-off between energy consumption and performance.
The relation between the energy consumption and the execution time for an
\end{equation}
%
where $Tnew$ is computed as in (\ref{eq:perf}) and $Told$ is computed as in (\ref{eq:told}).
-%
+\textcolor{blue}{
\begin{equation}
\label{eq:told}
- \Told = \mathop{\max_{i=1,2,\dots,N}}_{j=1,2,\dots,M} (\Tcp[ij]+\Tcm[ij])
+ \Told = \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}({\TcpOld[ij]} )
+ +\mathop{\min_{j=1,\dots,M_i}} (\Tcm[hj])
\end{equation}
-%
+}
In the same way, the energy is normalized by computing the ratio between the
consumed energy while scaling down the frequency and the consumed energy with
maximum frequency for all nodes:
%
\begin{equation}
\label{eq:eorginal}
- \Eoriginal = \sum_{i=1}^{N} \sum_{j=1}^{M} ( \Pd[ij] \cdot \Tcp[ij]) +
- \mathop{\sum_{i=1}^{N}} \sum_{j=1}^{M} (\Ps[ij] \cdot \Told)
+ \Eoriginal = \sum_{i=1}^{N} \sum_{j=1}^{M_i} ( \Pd[ij] \cdot \Tcp[ij]) +
+ \mathop{\sum_{i=1}^{N}} \sum_{j=1}^{M_i} (\Ps[ij] \cdot \Told)
\end{equation}
While the main goal is to optimize the energy and execution time at the same
\begin{equation}
\label{eq:max}
\MaxDist =
-\mathop{ \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}}_{k=1,\dots,F}
+\mathop{ \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}}_{k=1,\dots,F_j}
(\overbrace{\Pnorm(S_{ijk})}^{\text{Maximize}} -
\overbrace{\Enorm(S_{ijk})}^{\text{Minimize}} )
\end{equation}
-where $N$ is the number of clusters, $M$ is the number of nodes in each cluster and
-$F$ is the number of available frequencies for each node. Then, the optimal set
+where $N$ is the number of clusters, $M_i$ is the number of nodes in the cluster $i$ and
+$F_j$ is the number of available frequencies in the node $j$. Then, the optimal set
of scaling factors that satisfies (\ref{eq:max}) can be selected.
The objective function can work with any energy model or any power
values for each node (static and dynamic powers). However, the most important
is presented. It selects the vector of the frequency
scaling factors that gives the best trade-off between minimizing the
energy consumption and maximizing the performance of a message passing
-synchronous iterative application executed on a grid. It works
-online during the execution time of the iterative message passing program. It
+synchronous application \textcolor{blue}{with iterations} executed on a grid. It works
+online during the execution time of the message passing program \textcolor{blue}{with iterations}. It
uses information gathered during the first iteration such as the computation
time and the communication time in one iteration for each node. The algorithm is
executed after the first iteration and returns a vector of optimal frequency
program applies DVFS operations to change the frequencies of the CPUs according
to the computed scaling factors. This algorithm is called just once during the
execution of the program. Algorithm~\ref{dvfs} shows where and when the proposed
-scaling algorithm is called in the iterative MPI program.
+scaling algorithm is called in the MPI program \textcolor{blue}{with iterations}.
\begin{figure}[!t]
\centering
\includegraphics[scale=0.6]{fig/init_freq}
- \caption{Selecting the initial frequencies}
+ \caption{Selecting the initial frequencies in a grid platform}
\label{fig:st_freq}
\end{figure}
-Nodes from distinct clusters in a grid have different computing powers, thus
-while executing message passing iterative synchronous applications, fast nodes
-have to wait for the slower ones to finish their computations before being able
-to synchronously communicate with them as in Figure~\ref{fig:heter}. These
-periods are called idle or slack times. The algorithm takes into account this
+
+
+The algorithm takes into account this
problem and tries to reduce these slack times when selecting the vector of the frequency
scaling factors. At first, it selects initial frequency scaling factors
that increase the execution times of fast nodes and minimize the differences
of the slowest node and the computation time of the node $i$ as follows:
\begin{equation}
\label{eq:Scp}
- \Scp[ij] = \frac{ \mathop{\max\limits_{i=1,\dots N}}\limits_{j=1,\dots,M}(\Tcp[ij])} {\Tcp[ij]}
+ \Scp[ij] = \frac{ \mathop{\max\limits_{i=1,\dots N}}\limits_{j=1,\dots,M_i}(\Tcp[ij])} {\Tcp[ij]}
\end{equation}
Using the initial frequency scaling factors computed in (\ref{eq:Scp}), the
algorithm computes the initial frequencies for all nodes as a ratio between the
follows:
\begin{equation}
\label{eq:Fint}
- F_{ij} = \frac{\Fmax[ij]}{\Scp[ij]},~{i=1,2,\dots,N},~{j=1,\dots,M}
+ F_{ij} = \frac{\Fmax[ij]}{\Scp[ij]},~{i=1,2,\dots,N},~{j=1,\dots,M_i}
\end{equation}
If the computed initial frequency for a node is not available in the gears of
that node, it is replaced by the nearest available frequency. In
bound until all nodes reach their minimum frequencies or their lower bounds, to compute the overall
energy consumption and performance and selects the optimal vector of the frequency scaling
factors. At each iteration the algorithm determines the slowest node
-according to Equation~\ref{eq:perf}
-%\AG[]{Be consistent: remove word ``Equation'' and add parentheses around equation number, here and all along the rest of the text.}
-and keeps its frequency unchanged,
+according to Equation~\ref{eq:perf} and keeps its frequency unchanged,
while it lowers the frequency of all other nodes by one gear. The new overall
energy consumption and execution time are computed according to the new scaling
factors. The optimal set of frequency scaling factors is the set that gives the
consumed energy decrease from the beginning of the plot. On the other hand, in
the grid platform the performance is maintained at the beginning of the
plot even if the frequencies of the faster nodes decrease until the computing
-power of scaled down nodes are lower than the slowest node. In other words,
-\AG[]{That's not a sentence.}
-until they reach the higher bound. It can also be noticed that the higher the
-difference between the faster nodes and the slower nodes is, the bigger the
-maximum distance between the energy curve and the performance curve is, which results in bigger energy savings.
+power of scaled down nodes are lower than the slowest node. It can also be noticed that the higher the difference between the faster nodes and the slower nodes is, the bigger the maximum distance between the energy curve and the performance curve is, which results in bigger energy savings.
\section{Experimental results}
Two types of local networks are used, Ethernet or Infiniband networks which have different characteristics in terms of bandwidth and latency.
Since Grid'5000 is dedicated to testing, contrary to production grids it allows a user to deploy its own customized operating system on all the booked nodes. The user could have root rights and thus apply DVFS operations while executing a distributed application. Moreover, the Grid'5000 testbed provides at some sites a power measurement tool to capture
-the power consumption for each node in those sites. The measured power is the overall consumed power by all the components of a node at a given instant, such as CPU, hard drive, main-board, memory, \dots{} For more details refer to
+the power consumption for each node in those sites. The measured power is the overall consumed power by all the components of a node at a given instant, such as CPU, hard drive, main-board and memory. For more details refer to
\cite{Energy_measurement}. In order to correctly measure the CPU power of one core in a node $j$,
firstly, the power consumed by the node while being idle at instant $y$, noted as $\Pidle[jy]$, was measured. Then, the power was measured while running a single thread benchmark with no communication (no idle time) over the same node with its CPU scaled to the maximum available frequency. The latter power measured at time $x$ with maximum frequency for one core of node $j$ is noted $\Pmax[jx]$. The difference between the two measured power consumptions represents the
dynamic power consumption of that core with the maximum frequency, see Figure~\ref{fig:power_cons}.
\begin{figure}[!t]
\centering
\includegraphics[scale=0.6]{fig/power_consumption.pdf}
- \AG{I don't understand the labels on the horizontal axis: 10:30:37, 10:30:38,
- etc.}
\caption{The power consumption by one core from the Taurus cluster}
\label{fig:power_cons}
\end{figure}
The energy model and the scaling factors selection algorithm were applied to the NAS parallel benchmarks v3.3 \cite{NAS.Parallel.Benchmarks} and evaluated over Grid'5000.
-The benchmark suite contains seven applications: CG, MG, EP, LU, BT, SP and FT. These applications have different computations and communications ratios and strategies which make them good testbed applications to evaluate the proposed algorithm and energy model.
+The benchmark suite contains seven applications: CG, MG, EP, LU, BT, SP and FT. \textcolor{blue}{These benchmarks are message passing applications with iterations compute
+the same block of operations several times, starting from the initial solution until reaching
+the acceptable approximation of the exact solution.}
+ These applications have different computations and communications ratios and strategies which make them good testbed applications to evaluate the proposed algorithm and energy model.
The benchmarks have seven different classes, S, W, A, B, C, D and E, that represent the size of the problem that the method solves. In this work, class D was used for all benchmarks in all the experiments presented in the next sections.
\begin{table}[!t]
- \caption{CPUs characteristics of the selected clusters}
+ \caption{The characteristics of the CPUs in the selected clusters}
% title of Table
\centering
\begin{tabular}{|*{7}{c|}}
\subsection{The experimental results of the scaling algorithm}
\label{sec.res}
In this section, the results of the application of the scaling factors selection algorithm \ref{HSA}
-to the NAS parallel benchmarks are presented.
+to the NAS parallel benchmarks are presented. \textcolor{blue}{Each experiment of this section and next sections has been executed many times and the results presented in the figures are the average values of many execution.}
As mentioned previously, the experiments
were conducted over two sites of Grid'5000, Lyon and Nancy sites.
\begin{table}[h]
-\caption{The different clusters scenarios}
+\caption{The different grid scenarios}
\centering
\begin{tabular}{|*{4}{c|}}
\hline
\label{tab:sc}
\end{table}
-
-
+\begin{figure}
+ \centering
+ \subfloat[The energy consumption by the nodes wile executing the NAS benchmarks over different scenarios
+ ]{%
+ \includegraphics[width=.48\textwidth]{fig/eng_con_scenarios.eps}\label{fig:eng_sen}} \hspace{0.4cm}%
+ \subfloat[The execution times of the NAS benchmarks over different scenarios]{%
+ \includegraphics[width=.48\textwidth]{fig/time_scenarios.eps}\label{fig:time_sen}}
+ \label{fig:exp-time-energy}
+ \caption{The energy consumption and execution time of NAS Benchmarks over different scenarios}
+\end{figure}
The NAS parallel benchmarks are executed over these two platforms
with different number of nodes, as in Table~\ref{tab:sc}.
using the proposed frequency selection algorithm is measured
using the equation of the reduced energy consumption, Equation~\ref{eq:energy}. This model uses the measured dynamic power showed in Table~\ref{table:grid5000}
and the static
-power is assumed to be equal to 20\% of the dynamic power. The execution
+power is assumed to be equal to 20\% of the dynamic power \textcolor{blue}{as in \cite{Rauber_Analytical.Modeling.for.Energy}}. The execution
time is measured for all the benchmarks over these different scenarios.
The energy consumptions and the execution times for all the benchmarks are
The long distance communications between the two distributed sites increase the idle time, which leads to more static energy consumption.
The execution times of these benchmarks
-over one site with 16 and 32 nodes are also lower when compared to those of the two sites
-scenario. Moreover, most of the benchmarks running over the one site scenario their execution times are approximately divided by two when the number of computing nodes is doubled from 16 to 32 nodes (linear speed up according to the number of the nodes).\AG{Parse error (cannot understand the previous sentence).}
+over one site with 16 and 32 nodes are also lower than those of the two sites
+scenario. Moreover, most of the benchmarks running over the one site scenario have their execution times approximately halved when the number of computing nodes is doubled from 16 to 32 nodes (linear speed up according to the number of the nodes).
-However, the execution times and the energy consumptions of EP and MG
+However, the execution times and the energy consumptions of the EP and MG
benchmarks, which have no or small communications, are not significantly
affected in both scenarios, even when the number of nodes is doubled. On the
-other hand, the communications\AG[]{the communication time?} of the rest of the benchmarks increases when
-using long distance communications between two sites or increasing the number of
+other hand, the communication times of the rest of the benchmarks increase when
+using long distance communications between two sites or when increasing the number of
computing nodes.
-
The energy saving percentage is computed as the ratio between the reduced
energy consumption, Equation~\ref{eq:energy}, and the original energy consumption,
Equation~\ref{eq:eorginal}, for all benchmarks as in Figure~\ref{fig:eng_s}.
is exponentially related to the CPU's frequency value. On the other hand, the increase in the number of computing nodes can
increase the communication times and thus produces less energy saving depending on the
benchmarks being executed. The results of benchmarks CG, MG, BT and FT show more
-energy saving percentage in one site scenario when executed over 16 nodes comparing to 32 nodes. While, LU and SP consume more energy with 16 nodes than 32 in one site because their computations to communications ratio is not affected by the increase of the number of local communications.
-\begin{figure}
- \centering
- \subfloat[The energy consumption by the nodes wile executing the NAS benchmarks over different scenarios
- ]{%
- \includegraphics[width=.48\textwidth]{fig/eng_con_scenarios.eps}\label{fig:eng_sen}} \hspace{0.4cm}%
- \subfloat[The execution times of the NAS benchmarks over different scenarios]{%
- \includegraphics[width=.48\textwidth]{fig/time_scenarios.eps}\label{fig:time_sen}}
- \label{fig:exp-time-energy}
- \caption{The energy consumption and execution time of NAS Benchmarks over different scenarios}
-\end{figure}
+energy saving percentage in the one site scenario when executed over 16 nodes than over 32 nodes. LU and SP consume more energy with 16 nodes than 32 in one site because their computations to communications ratio is not affected by the increase of the number of local communications.
+\begin{figure*}[!h]
+ \centering
+ \subfloat[The energy reduction while executing the NAS benchmarks over different scenarios ]{%
+ \includegraphics[width=.48\textwidth]{fig/eng_s.eps}\label{fig:eng_s}} \hspace{0.4cm}%
+ \subfloat[The performance degradation of the NAS benchmarks over different scenarios]{%
+ \includegraphics[width=.48\textwidth]{fig/per_d.eps}\label{fig:per_d}}\hspace{0.4cm}%
+ \subfloat[The trade-off distance between the energy reduction and the performance of the NAS benchmarks
+ over different scenarios]{%
+ \includegraphics[width=.48\textwidth]{fig/dist.eps}\label{fig:dist}}
+ \label{fig:exp-res}
+ \caption{The experimental results of different scenarios}
+\end{figure*}
produces less energy consumption and thus more energy saving.
The best energy saving percentage was obtained in the one site scenario with 16 nodes, the energy consumption was on average reduced up to 30\%.
-\begin{figure*}[t]
- \centering
- \subfloat[The energy reduction while executing the NAS benchmarks over different scenarios ]{%
- \includegraphics[width=.48\textwidth]{fig/eng_s.eps}\label{fig:eng_s}} \hspace{0.4cm}%
- \subfloat[The performance degradation of the NAS benchmarks over different scenarios]{%
- \includegraphics[width=.48\textwidth]{fig/per_d.eps}\label{fig:per_d}}\hspace{0.4cm}%
- \subfloat[The trade-off distance between the energy reduction and the performance of the NAS benchmarks
- over different scenarios]{%
- \includegraphics[width=.48\textwidth]{fig/dist.eps}\label{fig:dist}}
- \label{fig:exp-res}
- \caption{The experimental results of different scenarios}
-\end{figure*}
+
Figure \ref{fig:per_d} presents the performance degradation percentages for all benchmarks over the two scenarios.
The performance degradation percentage for the benchmarks running on two sites with
16 or 32 nodes is on average equal to 8.3\% or 4.7\% respectively.
For this scenario, the proposed scaling algorithm selects smaller frequencies for the executions with 32 nodes without significantly degrading their performance because the communication times are higher with 32 nodes which results in smaller computations to communications ratio. On the other hand, the performance degradation percentage for the benchmarks running on one site with
-16 or 32 nodes is on average equal to 3.2\% or 10.6\% respectively. In contrary to the two sites scenario, when the number of computing nodes is increased in the one site scenario, the performance degradation percentage is increased. Therefore, doubling the number of computing
+16 or 32 nodes is on average equal to 3.2\% and 10.6\% respectively. In contrary to the two sites scenario, when the number of computing nodes is increased in the one site scenario, the performance degradation percentage is increased. Therefore, doubling the number of computing
nodes when the communications occur in high speed network does not decrease the computations to
communication ratio.
The performance degradation percentage of the EP benchmark after applying the scaling factors selection algorithm is the highest in comparison to
the other benchmarks. Indeed, in the EP benchmark, there are no communication and slack times and its
performance degradation percentage only depends on the frequencies values selected by the algorithm for the computing nodes.
-The rest of the benchmarks showed different performance degradation percentages, which decrease
+The rest of the benchmarks showed different performance degradation percentages which decrease
when the communication times increase and vice versa.
Figure \ref{fig:dist} presents the distance percentage between the energy saving and the performance degradation for each benchmark over both scenarios. The trade-off distance percentage can be
-\subsection{The experimental results over multi-cores clusters}
+\subsection{The experimental results over multi-core clusters}
\label{sec.res-mc}
The clusters of Grid'5000 have different number of cores embedded in their nodes
as shown in Table~\ref{table:grid5000}. In
this section, the proposed scaling algorithm is evaluated over the Grid'5000 platform while using multi-cores nodes selected according to the one site scenario described in Section~\ref{sec.res}.
-The one site scenario uses 32 cores from multi-cores nodes instead of 32 distinct nodes. For example if
+The one site scenario uses 32 cores from multi-core nodes instead of 32 distinct nodes. For example if
the participating number of cores from a certain cluster is equal to 14,
-in the multi-core scenario the selected nodes is equal to 4 nodes while using
-3 or 4 cores from each node. The platforms with one
-core per node and multi-cores nodes are shown in Table~\ref{table:sen-mc}.
-The energy consumptions and execution times of running class D of the NAS parallel
-benchmarks over these two different scenarios are presented
+in the multi-core scenario 4 nodes are selected and
+3 or 4 cores from each node are used. The platforms with one
+core per node and multi-core nodes are shown in Table~\ref{table:sen-mc}.
+The energy consumptions and execution times of running the class D of the NAS parallel
+benchmarks over these two different platforms are presented
in Figures \ref{fig:eng-cons-mc} and \ref{fig:time-mc} respectively.
-\begin{table}[]
+\begin{table}[!h]
\centering
-\caption{The multicores scenarios}
+\caption{The multi-core scenarios}
\begin{tabular}{|*{4}{c|}}
\hline
Scenario name & Cluster name & Nodes per cluster &
\multirow{3}{*}{One core per node} & Graphite & 4 & 1 \\ \cline{2-4}
& Graphene & 14 & 1 \\ \cline{2-4}
& Griffon & 14 & 1 \\ \hline
-\multirow{3}{*}{Multi-cores per node} & Graphite & 1 & 4 \\ \cline{2-4}
+\multirow{3}{*}{Multi-core per node} & Graphite & 1 & 4 \\ \cline{2-4}
& Graphene & 4 & 3 or 4 \\ \cline{2-4}
& Griffon & 4 & 3 or 4 \\ \hline
\end{tabular}
\end{table}
-\begin{figure}
+\begin{figure}[!h]
\centering
- \subfloat[Comparing the execution times of running NAS benchmarks over one core and multicores scenarios]{%
+ \subfloat[Comparing the execution times of running the NAS benchmarks over one core and multi-core scenarios]{%
\includegraphics[width=.48\textwidth]{fig/time.eps}\label{fig:time-mc}} \hspace{0.4cm}%
- \subfloat[Comparing the energy consumptions of running NAS benchmarks over one core and multi-cores scenarios]{%
+ \subfloat[Comparing the energy consumptions of running the NAS benchmarks over one core and multi-core scenarios]{%
\includegraphics[width=.48\textwidth]{fig/eng_con.eps}\label{fig:eng-cons-mc}}
\label{fig:eng-cons}
- \caption{The energy consumptions and execution times of NAS benchmarks over one core and multi-cores per node architectures}
+ \caption{The energy consumptions and execution times of the NAS benchmarks running over one core and multi-core per node architectures}
\end{figure}
-The execution times for most of the NAS benchmarks are higher over the multi-cores per node scenario
-than over single core per node scenario. Indeed,
- the communication times are higher in the one site multi-cores scenario than in the latter scenario because all the cores of a node share the same node network link which can be saturated when running communication bound applications. Moreover, the cores of a node share the memory bus which can be also saturated and become a bottleneck.
+The execution times for most of the NAS benchmarks are higher over the multi-core per node scenario
+than over the single core per node scenario. Indeed,
+ the communication times are higher in the one site multi-core scenario than in the latter scenario because all the cores of a node share the same node network link which can be saturated when running communication bound applications. Moreover, the cores of a node share the memory bus which can be also saturated and become a bottleneck.
Moreover, the energy consumptions of the NAS benchmarks are lower over the
- one core scenario than over the multi-cores scenario because
+ one core scenario than over the multi-core scenario because
the first scenario had less execution time than the latter which results in less static energy being consumed.
The computations to communications ratios of the NAS benchmarks are higher over
-the one site one core scenario when compared to the ratio of the multi-cores scenario.
+the one site one core scenario when compared to the ratio of the multi-core scenario.
More energy reduction can be gained when this ratio is big because it pushes the proposed scaling algorithm to select smaller frequencies that decrease the dynamic power consumption. These experiments also showed that the energy
consumption and the execution times of the EP and MG benchmarks do not change significantly over these two
scenarios because there are no or small communications. Contrary to EP and MG, the energy consumptions and the execution times of the rest of the benchmarks vary according to the communication times that are different from one scenario to the other.
\begin{figure*}[t]
\centering
- \subfloat[The energy saving of running NAS benchmarks over one core and multicores scenarios]{%
+ \subfloat[The energy saving of running NAS benchmarks over one core and multicore scenarios]{%
\includegraphics[width=.48\textwidth]{fig/eng_s_mc.eps}\label{fig:eng-s-mc}} \hspace{0.4cm}%
- \subfloat[The performance degradation of running NAS benchmarks over one core and multicores scenarios
+ \subfloat[The performance degradation of running NAS benchmarks over one core and multi-core scenarios
]{%
\includegraphics[width=.48\textwidth]{fig/per_d_mc.eps}\label{fig:per-d-mc}}\hspace{0.4cm}%
- \subfloat[The trade-off distance of running NAS benchmarks over one core and multicores scenarios]{%
+ \subfloat[The trade-off distance of running NAS benchmarks over one core and multicore scenarios]{%
\includegraphics[width=.48\textwidth]{fig/dist_mc.eps}\label{fig:dist-mc}}
\label{fig:exp-res2}
- \caption{The experimental results of one core and multi-cores scenarios}
+ \caption{The experimental results of one core and multi-core scenarios}
\end{figure*}
-The energy saving percentages of all NAS benchmarks running over these two scenarios are presented in Figure~\ref{fig:eng-s-mc}.
+The energy saving percentages of all the NAS benchmarks running over these two scenarios are presented in Figure~\ref{fig:eng-s-mc}.
The figure shows that the energy saving percentages in the one
-core and the multi-cores scenarios
+core and the multi-core scenarios
are approximately equivalent, on average they are equal to 25.9\% and 25.1\% respectively.
The energy consumption is reduced at the same rate in the two scenarios when compared to the energy consumption of the executions without DVFS.
The performance degradation percentages of the NAS benchmarks are presented in
-Figure~\ref{fig:per-d-mc}. It shows that the performance degradation percentages are higher for the NAS benchmarks over the one core per node scenario (on average equal to 10.6\%) than over the multi-cores scenario (on average equal to 7.5\%). The performance degradation percentages over the multi-cores scenario are lower because the computations to communications ratios are smaller than the ratios of the other scenario.
+Figure~\ref{fig:per-d-mc}. It shows that the performance degradation percentages are higher for the NAS benchmarks executed over the one core per node scenario (on average equal to 10.6\%) than over the multi-core scenario (on average equal to 7.5\%). The performance degradation percentages over the multi-core scenario are lower because the computations to communications ratios are smaller than the ratios of the other scenario.
-The trade-off distances percentages of the NAS benchmarks over the two scenarios are presented
-in ~Figure~\ref{fig:dist-mc}. These trade-off distances between energy consumption reduction and performance are used to verify which scenario is the best in both terms at the same time. The figure shows that the trade-off distance percentages are on average bigger over the multi-cores scenario (17.6\%) than over the one core per node scenario (15.3\%).
+The trade-off distances percentages of the NAS benchmarks over both scenarios are presented
+in ~Figure~\ref{fig:dist-mc}. These trade-off distances between energy consumption reduction and performance are used to verify which scenario is the best in both terms at the same time. The figure shows that the trade-off distance percentages are on average bigger over the multi-core scenario (17.6\%) than over the one core per node scenario (15.3\%).
The aim of this section is to evaluate the scaling algorithm while assuming different values of static powers.
In addition to the previously used percentage of static power, two new static power ratios, 10\% and 30\% of the measured dynamic power of the core, are used in this section.
The experiments have been executed with these two new static power scenarios over the one site one core per node scenario.
-In these experiments, class D of the NAS parallel benchmarks are executed over the Nancy site. 16 computing nodes from the three clusters, Graphite, Graphene and Griffon, where used in this experiment.
+In these experiments, the class D of the NAS parallel benchmarks were executed over the Nancy site. 16 computing nodes from the three clusters, Graphite, Graphene and Griffon, were used in this experiment.
\begin{figure*}[t]
\end{figure}
The energy saving percentages of the NAS benchmarks with the three static power scenarios are presented
-in Figure~\ref{fig:eng_sen}. This figure shows that the 10\% of static power scenario
+in Figure~\ref{fig:eng-pow}. This figure shows that the 10\% of static power scenario
gives the biggest energy saving percentages in comparison to the 20\% and 30\% static power
scenarios. The small value of the static power consumption makes the proposed
scaling algorithm select smaller frequencies for the CPUs.
These smaller frequencies reduce the dynamic energy consumption more than increasing the consumed static energy which gives less overall energy consumption.
-The energy saving percentages of the 30\% static power scenario is the smallest between the other scenarios, because the scaling algorithm selects bigger frequencies for the CPUs which increases the energy consumption. Figure \ref{fig:fre-pow} demonstrates that the proposed scaling algorithm selects the best frequency scaling factors according to the static power consumption ratio being used.
+The energy saving percentages of the 30\% static power scenario is the smallest between the other scenarios, because the scaling algorithm selects bigger frequencies for the CPUs which increases the energy consumption. Figure \ref{fig:fre-pow} demonstrates that the proposed scaling algorithm selects the best frequency scaling factors according to the static power consumption ratio being used.
The performance degradation percentages are presented in Figure~\ref{fig:per-pow}.
The 30\% static power scenario had less performance degradation percentage because the scaling algorithm
had selected big frequencies for the CPUs. While,
the inverse happens in the 10\% and 20\% scenarios because the scaling algorithm had selected CPUs' frequencies smaller than those of the 30\% scenario. The trade-off distance percentage for the NAS benchmarks with these three static power scenarios
-are presented in Figure~\ref{fig:dist}.
+are presented in Figure~\ref{fig:dist-pow}.
It shows that the best trade-off
distance percentage is obtained with the 10\% static power scenario and this percentage
is decreased for the other two scenarios because the scaling algorithm had selected different frequencies according to the static power values.
Finding the frequencies that give the best trade-off between the energy consumption and the performance for a parallel
application is not a trivial task. Many algorithms have been proposed to tackle this problem.
In this section, the proposed frequencies selecting algorithm is compared to a method that uses the well known energy and delay product objective function, $EDP=energy \times delay$, that has been used by many researchers \cite{EDP_for_multi_processors,Energy_aware_application_scheduling,Exploring_Energy_Performance_TradeOffs}.
-This objective function was also used by Spiliopoulos et al. algorithm \cite{Spiliopoulos_Green.governors.Adaptive.DVFS} where they select the frequencies that minimize the EDP product and apply them with DVFS operations to the multi-cores
+This objective function was also used by Spiliopoulos et al. algorithm \cite{Spiliopoulos_Green.governors.Adaptive.DVFS} where they select the frequencies that minimize the EDP product and apply them with DVFS operations to the multi-core
architecture. Their online algorithm predicts the energy consumption and execution time of a processor before using the EDP method.
-
-To fairly compare the proposed frequencies scaling algorithm to Spiliopoulos et al. algorithm, called Maxdist and EDP respectively, both algorithms use the same energy model, Equation~\ref{eq:energy} and
-execution time model, Equation~\ref{eq:perf}, to predict the energy consumption and the execution time for each computing node.
-Moreover, both algorithms start the search space from the upper bound computed as in Equation~\ref{eq:Fint}.
-Finally, the resulting EDP algorithm is an exhaustive search algorithm that tests all the possible frequencies, starting from the initial frequencies (upper bound),
-and selects the vector of frequencies that minimize the EDP product.
-
-Both algorithms were applied to class D of the NAS benchmarks over 16 nodes.
-The participating computing nodes are distributed according to the two scenarios described in Section~\ref{sec.res}.
-The experimental results, the energy saving, performance degradation and trade-off distance percentages, are
-presented in Figures~\ref{fig:edp-eng}, \ref{fig:edp-perf} and \ref{fig:edp-dist} respectively.
-
-
\begin{figure*}[t]
\centering
\subfloat[The energy reduction induced by the Maxdist method and the EDP method]{%
\label{fig:edp-comparison}
\caption{The comparison results}
\end{figure*}
+To fairly compare the proposed frequencies scaling algorithm to Spiliopoulos et al. algorithm, called Maxdist and EDP respectively, both algorithms use the same energy model, Equation~\ref{eq:energy} and
+execution time model, Equation~\ref{eq:perf}, to predict the energy consumption and the execution time for each computing node.
+Moreover, both algorithms start the search space from the upper bound computed as in Equation~\ref{eq:Fint}.
+Finally, the resulting EDP algorithm is an exhaustive search algorithm that tests all the possible frequencies, starting from the initial frequencies (upper bound),
+and selects the vector of frequencies that minimize the EDP product.
+
+Both algorithms were applied to the class D of the NAS benchmarks running over 16 nodes.
+The participating computing nodes are distributed according to the two scenarios described in Section~\ref{sec.res}.
+The experimental results, the energy saving, performance degradation and trade-off distance percentages, are
+presented in Figures~\ref{fig:edp-eng}, \ref{fig:edp-perf} and \ref{fig:edp-dist} respectively.
As shown in these figures, the proposed frequencies selection algorithm, Maxdist, outperforms the EDP algorithm in terms of energy consumption reduction and performance for all of the benchmarks executed over the two scenarios.
-The proposed algorithm gives better results than EDP because it
+The proposed algorithm gives better results than the EDP method because it
maximizes the energy saving and the performance at the same time.
Moreover, the proposed scaling algorithm gives the same weight for these two metrics.
Whereas, the EDP algorithm gives sometimes negative trade-off values for some benchmarks in the two sites scenarios.
These negative trade-off values mean that the performance degradation percentage is higher than the energy saving percentage.
The high positive values of the trade-off distance percentage mean that the energy saving percentage is much higher than the performance degradation percentage.
-The time complexity of both Maxdist and EDP algorithms are $O(N \cdot M \cdot F)$ and
-$O(N \cdot M \cdot F^2)$ respectively, where $N$ is the number of the clusters, $M$ is the number of nodes and $F$ is the
-maximum number of available frequencies. When Maxdist is applied to a benchmark that is being executed over 32 nodes distributed between Nancy and Lyon sites, it takes on average $0.01 ms$ to compute the best frequencies while EDP is on average ten times slower over the same architecture.
+The complexity of both algorithms, Maxdist and EDP, are of order $O(N \cdot M_i \cdot F_j)$ and
+$O(N \cdot M_i \cdot F_j^2)$ respectively, where $N$ is the number of the clusters, $M_i$ is the number of nodes and $F_j$ is the
+maximum number of available frequencies. When Maxdist is applied to a benchmark that is being executed over 32 nodes distributed between Nancy and Lyon sites, it takes on average $0.01$ $ms$ to compute the best frequencies while the EDP method is on average ten times slower over the same architecture.
\section{Conclusion}
The algorithm selects the best vector of
frequencies that maximizes the trade-off distance
between the predicted energy consumption and the predicted execution time of the distributed
-iterative applications running over a heterogeneous grid. A new energy model
+ applications \textcolor{blue}{with iterations} running over a heterogeneous grid. A new energy model
is used by the proposed algorithm to predict the energy consumption
-of the distributed iterative message passing application running over a grid architecture.
+of the distributed message passing application \textcolor{blue}{with iterations} running over a grid architecture.
To evaluate the proposed method on a real heterogeneous grid platform, it was applied on the
- NAS parallel benchmarks and the class D instance was executed over the Grid'5000 testbed platform.
- The experiments on 16 nodes, distributed over three clusters, showed that the algorithm on average reduces by 30\% the energy consumption
+NAS parallel benchmarks and the class D instance was executed over the Grid'5000 testbed platform.
+The experiments executed on 16 nodes, distributed over three clusters, showed that the algorithm on average reduces by 30\% the energy consumption
for all the NAS benchmarks while on average only degrading by 3.2\% the performance.
-The Maxdist algorithm was also evaluated in different scenarios that vary in the distribution of the computing nodes between different clusters' sites or use multi-cores per node architecture or consume different static power values. The algorithm selects different vectors of frequencies according to the
+The Maxdist algorithm was also evaluated in different scenarios that vary in the distribution of the computing nodes between different clusters' sites or use multi-core per node architecture or consume different static power values. The algorithm selects different vectors of frequencies according to the
computations and communication times ratios, and the values of the static and measured dynamic powers of the CPUs.
Finally, the proposed algorithm was compared to another method that uses
the well known energy and delay product as an objective function. The comparison results showed
that the proposed algorithm outperforms the latter by selecting a vector of frequencies that gives a better trade-off between energy consumption reduction and performance.
-In the near future, we would like to develop a similar method that is adapted to
-asynchronous iterative applications where iterations are not synchronized and communications are overlapped with computations.
+In the near future, \textcolor{blue}{we will adapt the proposed algorithm to take the variability between some iterations in two steps. In the first step, the algorithm selects the best frequencies at the end of the first iterations and apply them to the system. In the second step, after some iterations (e.g. 5 iterations) the algorithm recomputes the frequencies depending on the average of the communication and computation times for all previous iterations. It will change the frequency of each node if the new frequency is different from the old one. Otherwise, it keeps the old frequency.}
+Also, we would like to develop a similar method that is adapted to
+asynchronous applications \textcolor{blue}{with iterations} where iterations are not synchronized and communications are overlapped with computations.
The development of such a method might require a new energy model because the
number of iterations is not known in advance and depends on
the global convergence of the iterative system.
